Lecture 3 How to value bonds and
common stocks
Corporate Finance
Lecturer: Quan, Qi
Spring 2010
Homework for lecture 2
1. As winner of a breakfast cereal competition, you can
choose one of the following prizes:
 a. $100,000 now.
 b. $180,000 at the end of five years.
 c. $11,400 a year forever.
 d. $19,000 for each of 10 years.
 e. $6,500 next year and increasing thereafter by 5 percent
a year forever.
If the interest rate is 12 percent, which is the most valuable
prize?
2
Solution
a. PV = $100,000
b. PV = $180,000/(1+0.12)5 = $102,137
c. A perpetuity: PV = $11,400/0.12 = $95,000 (a perpetuity)
d. An annuity: PV = $19,000  [Annuity factor, 12%, t = 10]
= $19,000  5.650 = $107,350
e. A growing perpetuity:
PV = $6,500/(0.12 - 0.05) = $92,857
Prize (d) is the winner with the highest present value
3
Question 2
You are considering the purchase of an apartment complex that will generate a
net cash flow of $400,000 per year. You normally demand a 10 percent rate of
return on such investments. Future cash flows are expected to grow with
inflation at 4 percent per year. How much would you be willing to pay for the
complex if it:
 a. Will produce cash flows forever?
 b. Will have to be torn down in 20 years? Assume that the site will be worth
$5 million at that time net of demolition costs. (The $5 million includes 20
years’ inflation.)
Now calculate the real discount rate corresponding to the 10 percent nominal rate.
Redo the calculations for parts (a) and (b) using real cash flows. (Your answers
should not change.)
4
Solution to question 2
With the nominal cash flows:
a. Given the inflation rate of 4%, the cash flow in 1 year will be $416,000
and:
PV = $416,000/(0.10 - 0.04) = $6,933,333
b. The nominal cash flows are growing annuity for 20 years, with an
additional payment of $5 million at year 20:
 416,000
432,640
876,449
5,000,000 
PV  

. . . 

  $5,418,389
1
2
20
(1.10)
(1.10)
(1.10)20 
 (1.10)
5
Second, with real cash flows:
The real rate (r) is (1 + 0.10) = (1 + r)  (1.04)  r = 0.0577 = 5.77%
a.
PV = $400,000/(0.0577) = $6,932,409
b.
Now, the real cash flows are $400,000 per year for 20 years and $5
million (nominal) in 20 years. In real terms, the $5 million dollar
payment is:
$5,000,000/(1.04)20 = $2,281,935
Thus, the present value of the project is:

 $2,281,935
1
1
PV  $400,000  


 $5,417,986
20 
20
(0.0577)
(0.0577)(1.0577)
(1.0577)


6
Topics Covered

How to value bonds

How to value common stocks
7
How to value bonds

What is a bond
– A certificate showing that a borrower owes a specified sum; the
borrower agrees to make interest and principal payments on
designated dates
–
Components of bonds: face value, coupon rate, maturity, how
often the coupons are paid (annually or semiannually?)
8
9
How to value bonds- Zero-coupon bond

Pure discount bond (zero-coupon
bond): promises a single payment at a
fixed future date
–
If the payment is one (two) year (s)
from now, it is called a one-year (two
year) discount bond

What is the PV of a zero-coupon bond
that pays a face value of F in T years

Value of a zero-coupon bond

Example: A bond with a face value of
100 that matures in 20 years. Suppose
the interest rate is 10%
PV 
PV 
F
(1  r )T
100
20  14.86
(1.1)
10
How to value bonds- Level-coupon bond

What are level-coupon bonds
–
These bonds offer cash payments not just at maturity, but also at regular
times in between (e.g. payments on U.S. government issues and
American corporate bonds are made every six months until the bond
matures)
–
Example: There is a 5.375 percent bond with $100 face value maturing in
January 2016. You receive an interest payment of 0.05375*100=$5.375,
which is the bond’s coupon each year until 2016. When the bond matures
in 2016 the government pays you the final $5.375 interest, plus the $100
face value.
11
Cash Flows ($)
2011
2012
2013
2014
2015
2016
5.375
5.375
5.375
5.375
5.375
105.375
PV= PV (coupon payments) + PV (final payment)
= (coupon*6-year annuity factor) + (final payment * discount factor)
=5.375 [1/.038-1/0.38(1.038)6 ]+ 100/(1+0.038)6 = 28.36+79.95=108.31
12
How to value bonds – Yield to maturity

If the price of the bond is 108.31, what return do investors expect?
5.375 5.375
5.375
5.375
5.375 105.375
108.31 





2
3
4
5
1  r (1  r ) (1  r ) (1  r) (1  r)
(1  r)6

This r is called the bond’s yield to maturity

How to find the yield to maturity? By trial and error
– What if the yield falls to 2%? The price of the bond would go up to
118.90
– What if the yield rises to 10%? The price of the bond would fall to
79.86
13
How to value bonds – Semiannual coupons
and bond prices

A typical 6.25% U.S. Treasury bond of 2007 making coupon
payments semi-annually
– The bondholder gets 3.125 percent every half-year
– The sixth monthly payment is 0.03125*1000 (face value)=31.25

Suppose you could buy it on a yield of 2.16% in February 2004
– Since it is semiannually compounded, the yield over six months
should be 2.16/2=1.08%
– PV= 3.125  3.125  3.125  3.125  3.125  1003.125  1,118.19
1.0108
(1.0108)2
(1.0108)3
(1.0108)4
(1.0108)5
(1.0108)6
14
How to value common stocks
15
How to value common stocks

The discounted-cash-flow (DCF) formula for the present value of a
stock is the same as that of any other asset

What do shareholders receive from the company?
–
–

A stream of dividends
PV(stock) = PV(expected future dividends)
Why does the formula say nothing about capital gains?
16
How to value common stocks

What determines today’s price?
–
The cash payoff to owners of common stocks comes in two forms

–
1) cash dividends 2) capital gains
The current price of a share is P0;the expected price at the end of a year is
P1; the expected dividend per share is DIV1
Expected return= r =
Div1  P1  P 0
P0
This expected return is often called the market capitalization rate
If the P0=100, Div1=5 and P1=110, we have r=15%
17
How to value common stocks

You can also predict today’s price like this
Price=P0=

DIV 1  P1
1 r
How do we decide on r?
–
All securities in an equivalent risk class are priced to offer the same
expected return
–
P0=(5+100)/(1+0.15)=100
What if today’s price is higher (lower) than 100?
18
How to value common stocks

What determines next year’s price?
–
P1 =
Div 2  P 2
1 r
19
How to value common stocks

How far can we look if we continue working like this?

As time goes by, the dividend stream accounts for an
increasing proportion of present value. If the horizon is
infinite, we have
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Estimating the cost of equity capital


Suppose the expected dividends grow at a constant rate
What does that remind you of?
–
–

Like a growing perpetuity
Constant-growth DCF formula P0 =
DIV 1
rg
Cautions about using this formula: g must be less than r; g
cannot be abnormally high forever
21
Estimating the cost of equity capital

Can we use this growing perpetuity formula to derive r,
the discount rate (market capitalization rate)?
r= DIV1/P0+g
There are two parts in the formula: Dividend yield (DIV1/P0) and the
expected rate of growth in dividends (g)
22
Estimating the cost of equity capital

Example: The Cascade Natural Gas company
The stock price is $22.35 at the beginning of 2004. Dividends
payments for the next year were expected to be $1.03 per share
First part of the formula: Dividend yield= DIV1/P0=1.03/22.35=4.6%
What about g?
Analysts were forecasting an annual growth of 5.7%. Given this
estimate, r=4.6%+5.7%=10.3%
23
Estimating the cost of equity capital



The alternative approach to estimating long-run growth with payout
ratio
– Payout ratio: the ratio of dividends to earnings per share (EPS)
– If the payout ratio for Cascade is 66%, then the plowback ratio
would be 1-66% = 44%
– If ROE = EPS/book equity per share = 12%
– Dividend growth rate = g = plowback ratio*ROE=.44*.12=.53
– r= DIV1/P0 + g = 4.6% + 5.3% = 9.9%
Different methods could give different answers
Normally we take an average of r for similar firms
24
DCF valuation with varying growth rates

Two-stage DCF valuation model
P0 = PV(first-stage
dividends) +
PV(second-stage
dividends)
25
The link between stock price and earnings
per share

Growth stocks vs. income stocks
–
–

Growth stocks: with low or even no dividends, you buy them for the
expectation of capital gains
Income stocks: you buy them primarily for the cash dividends
Imagine a company that does not grow at all. All the earnings are paid
out as dividends
–
–
–
Expected return = dividend yield = earnings-price ratio
If the dividend is $10 and the price is $100, then the expected return
would be 10%
The company decides to invest $10 per share next year and the project
would earn $1 in addition in each subsequent year


NPV per share at year 1 = -10 + 1/0.1=0
The decision would have no impact on the stock price
26
The link between stock price and earnings
per share


The earnings-price ratio (EPS) equals the market capitalization rate (r) only when the new project’s
NPV=0
If the company reinvests earnings at below r, the earnings may increase but the share value is
reduced
27
Calculate the present value of growth
opportunities

In general, we can think of stock price as the capitalized value of
average earnings under a no-growth policy, plus PVGO, the net
present value of growth opportunities
EPS 1
P0 
 PVGO
R

EPS
PVGO
 r (1 
)
P0
P0
Example: how to value a growth stock using this model?
28
Calculate the present value of growth
opportunities

Suppose A company’s market capitalization rate, r, is 15%
and it is expected to pay a dividend of $5 in the first year.
Also, the constant growth rate of the dividend is 10%
29
Calculate the present value of growth
opportunities

Suppose the earnings per share is 8.33
If there is a no-growth policy, the capitalized value per share would be
EPS1/r=8.33/0.15=55.66
How to account for the difference between 55.66 and 100?
–
–
–
–
–
–
–
Payout ratio= DIV1/EPS1=5/8.33=.6, suppose ROE=25%
Growth rate = g =plowback ratio*ROE=0.4*25%=10%
The cash generated by this investment is 0.25*3.33=0.83
NPV1= -3.33+0.83/0.15=2.22; NPV2=-3.67+(0.83*1.10)/0.15=2.44
Present value of growth opportunities=PVGO=NPV1/(r-g)=44.44
Share price= present value of level stream of earnings + present value of
growth opportunities=55.66+44.44
30
Practice question
Pigeon express currently plows back 40 percent of its earnings and earns a return
of 20 percent on this investment. The dividend yield on the stock is 4 percent.
a. Assuming that Crecimiento can continue to plow back this proportion of
earnings and earn a 20 percent return on the investment, how rapidly will
earnings and dividends grow? What is the expected return on Crecimiento
stock?
b. Suppose that management suddenly announces that future investment
opportunities have dried up. Now Crecimiento intends to pay out all its earnings.
How will the stock price change?
c. Suppose that management simply announces that the expected return on new
investment would in the future be the same as the market capitalization rate. Now
what is Crecimiento’s stock price?
31
a. g = plowback ratio  ROE=0.40  0.20=8%
Using r = dividend yield + growth rate = 0.04 + 0.08 = 0.12 = 12.0%
b. Since the dividend yield = 4%, we have DIV1/P0 = 0.04 and DIV1=0.04*P0
A plowback ratio of 0.4 implies a payout ratio of 0.6, and we have
DIV1/EPS1 = 0.6
DIV1 = 0.6  EPS1
Equating these two expressions for DIV1 gives a relationship between price and
earnings per share:
0.04  P0 = 0.6  EPS1
EPS1/P0 = 0.066
 PVGO 
EPS 1
Also, we know that:
 r  1 

P0

P0

With (EPS1/P0) = 0.066 and r = 0.12, the ratio of the present value of growth opportunities
to price is 44.4 percent. Thus, if there are suddenly no future investment opportunities,
the stock price will decrease by 44.4 percent.
32